The Larson Research Center

As we’ve seen, the new scalar math requires a new scalar number line. The familiar number line, though simple and straightforward, is philosophically troublesome due to the enigmatic status of zero and negative numbers. Even so, it has been used to define integers and rational numbers, using a concept of 0 as a sort of number and -1 as the foundation of a set of multi-dimensional algebras called Lie algebras.

This has worked out fairly well for the research program of physics based on the vectorial motion in the LST, but it is totally unsuitable for the physics based on the scalar motion of the RST. We need a more complex, complete and consistent view of the scalar number line in order to use scalar mathematics in the development of the RST’s scalar theory.

However, the first thing we notice is that the RST’s scalar progression is 3D and therefore non-linear. Fortunately, though, we can use the combination of Larson’s 2x2x2 cube and its associated inner and outer spheres to construct a new, mult-dimensional, scalar number line that is linear. There are several aspects to this approach and, to understand it, we will have to take them one at a time.

The first thing we want to note is that the multi-dimensional magnitudes of the cube are integer indexes to the non-integer multi-dimensional magnitudes of the associated spheres. This is important to understand, since it enables us to unify the integer and non-integer magnitudes the way nature does, and, hopefully, it provides the key to understanding the mysterious connection between mathematics and physics.

To demonstrate what is meant by indexing the continuous magnitudes of physical variables with the discrete variables of numbers, we need to begin by analyzing the dimensions of Larson’s cube, as shown in figure 1 below.

Figure 1. Multi-Dimensional Number Line from Expansion of Larson’s 2x2x2 3D Cube.

In a 3D numerical progression, n3, all three dimensions (four counting 0) - the dimensional resolutes we might say - expand with the cube simultaneously. The magnitude of the 0 dimensional expansion, n(20), increases as a function of one-half of any given axis; the magnitude of the 1 dimensional expansion, 3(2n), increases as a function of the six 1D “directions” of the three axes; the magnitude of the 2 dimensional expansion, 3(2n)2, increases as a function of the 12 2D “directions” of the three axes, and the magnitude of the 3 dimensional expansion, (2n)3, increases as a function of the eight 3D “directions” of the three axes.

Figure 1 shows only one quadrant of the expanding cube, and the inner row/column is labeled with the 0 dimensional numbers, while the corresponding 1D, 2D and 3D numbers are labeled as successive outer layers of the quadrant (the factors of 3 in the 1D and 2D numbers comes from the 3 axes of expansion.)

By selecting just one quadrant of the expanding cube and labeling the magnitudes of all four dimensions in this manner, we get a scalar number line, where the vertical line is independent of the horizontal line, which will eventually allow us to include the reciprocal property that at this point is not apparent. In addition to assuming the presence of the other quadrants in the expansion, the figure does not show the third dimension graphically, but assumes its presence (the z axis with magnitudes in front of and behind the page.)

By accommodating all the magnitudes of the four dimensions this way, we can simplify the required graphics considerably, while maintaining the 3D scalar concept. Next, we can add the non-integer complement magnitudes of the associated inner and outer spheres to figure 1, as shown in figure 2 below.

Figure 2. Multi-Dimensional Number Line from Larson’s Cube with Inner and Outer Spheres

Of course, the magnitude of the radius of the inner sphere is always an integer and that of the outer sphere is always a non-integer, when n >= 1. Multiplying the multi-dimensional values of the outer radii by factors of pi, we obtain 1D (circumference), 2D (surface) and 3D (volume) continuous multi-dimensional magnitudes, indexed by the corresponding integers.

However, while the outer radii are multiples of the square root of 2, when the 0D magnitudes are greater than 1, the inner radii are inverse multiples of the inverse of the square root of 2, when the 0D magnitudes are less than 1, as shown in figures 3 and 4 below.

Figure 3. Outer Radii are Multiples of the Square Root of 2 at Indexes Greater Than 1.

Figure 4. Inner Radii are the Inverse Multiples of the Inverse of the Square Root of 2 at Indexes Less Than 1.

Hence, we can plot the radii linearly on a line, what we are want to call the new scalar number line:

…1/3(1/21/2), 1/2(1/21/2), 1/1(1/21/2), 1/1(2/11/2), 2/1(2/11/2), 3/1(2/11/2)…

comparing this to the traditional scalar number line:

…1/3, 1/2, 1/1, 2/1, 3/1…,

we see several differences. First, there is a distinct difference between the counting multiple and the unit. In the traditional line they are one and the same: 1(1), 2(1), 3(1), …, but in the new line the counting multiple, successive increments of 1, is very different from the unit, which is the square root of 2.

Proceeding in the opposite “direction,” the counting multiple of the traditional line is the inverse of the positive multiple, while the unit is the inverse of 1: …1/3(1/1), 1/2(1/1), 1/1(1/1), but because the inverse of 1/1 is indistinguishable from 1/1, it is not recognized that there are TWO units involved, where one is the inverse of the other.

In the new line, the unit of the outer sphere is the square root of 2, while the unit of the inner sphere is the inverse of the square root of 2, as can be clearly seen by comparing figures 3 and 4, so this requires two instances of the mathematical value of 1, if you will.

Interestingly enough, one of the confusing issues of working with the scalar concepts of the RST, is that while 1/1 is equal to 1/1 mathematically, s/t is not equal to t/s physically. The new scalar number line should be a great help in this regard.

Update: I just noticed that the graphic in figure 4 is the wrong one. I’m making a new graphic for it now and will update the figure soon.

 Update: Replaced graphic in figure 4 (please pardon the distortions.)